Singular Supports in Equal and Mixed Characteristics - 2/4

Explore advanced concepts in algebraic geometry through this comprehensive lecture on singular supports of constructible sheaves, delivered by Takeshi Saito from the University of Tokyo at the Institut des Hautes Etudes Scientifiques. Delve into Beilinson's definition of singular support as a closed conical subset on the cotangent bundle of smooth schemes over fields, examining its fundamental properties and existence proofs through Radon transform techniques. Learn about the Braverman-Gaitsgory interpretation of local acyclicity and understand how this provides an equivalent formulation of the original definition. Investigate the challenges and developments in mixed characteristic theory, where traditional approaches face limitations and new tools become necessary. Discover the introduction of the Frobenius-Witt cotangent bundle as a replacement for the standard cotangent bundle in mixed characteristics, understanding its correct rank properties despite being defined only on characteristic p fibers. Examine the definition of singular support and its relative variants in this context, and follow the adaptation of Beilinson's Radon transform argument to prove the existence of saturation for the relative variant. This lecture forms part of a series addressing both equal and mixed characteristic cases, providing essential insights into modern developments in the geometric theory of constructible sheaves.